\section{1.11} 
\begin{frame}[allowframebreaks]{1.11. }

\vspace{-0.4cm}

1.11. If $A$ is a category, then, as usual $D^b(A)$ (resp. $D^-(A)$, resp. $D^+(A)$) is the derived category of bounded (resp. bounded above, resp. bounded below) complexes in $A$. 

We shall however write $D^b(\mathcal{D}_X)$ for $D^b(\mu(\mathcal{D}_X))$ since this is the derived category of interest to us. 

Let us denote by $D^b_{qc}(\mathcal{D}_X)$ the derived category of bounded complexes $F^\bullet$ in $\mathrm{Mod}(\mathcal{D}_X)$ whose cohomology sheaves are quasi-coherent. 

If $F^\bullet$ consists of q.c. $\mathcal{D}_X$-modules, then obviously $H^iF^\bullet$ is also q.c. 

The natural inclusion $\mu(\mathcal{D}_X) \to \mathrm{Mod}(\mathcal{D}_X)$ induces therefore a morphism of derived categories:
\begin{equation}
\alpha: D^b(\mathcal{D}_X) \to D^b_{qc}(\mathrm{Mod}(\mathcal{D}_X))
\end{equation}

J. Bernstein has recently shown that $\alpha$ is an equivalence of categories. 

His proof (or a more general result in fact) will be given at the end of Section 2.

We let $\mathrm{coh}(\mathcal{D}_X)$ be the category of coherent $\mathcal{D}_X$-modules and $D^b_{\mathrm{coh}}(\mathcal{D}_X)$ the bounded derived category of q.c. $\mathcal{D}_X$-modules whose cohomology is coherent. 

A bounded complex of coherent $\mathcal{D}_X$-modules obviously belongs to it, whence again a natural morphism
\[
D^b(\mathrm{coh}(\mathcal{D}_X)) \to D^b_{\mathrm{coh}}(\mathcal{D}_X).
\]
It is also an equivalence of categories. 

Once it is known that any element of $\mu(\mathcal{D}_X)$ is an inductive limit of coherent submodules (2.3), this follows from an argument communicated to me by Deligne (which yields more generally an equivalence between the bounded derived category of an abelian category $A$ and the bounded derived category of the ind-objects of $A$ with cohomology in $A$). 

For the sake of completeness, the proof in the case of interest here is given at the end of § 2 (2.11).

\end{frame}

